Papers

We prove that, for every $n\ge1$, the degree-$n$ Bernstein operator on $[0,\pi]$ preserves positive-definiteness on the circle $S^1$. Equivalently, if a continuous function on $[0,\pi]$ defines a positive-definite isotropic kernel on $S^1$, then its Bernstein polynomial approximation of any fixed degree does as well. The proof reduces the problem to the nonnegativity of the cosine coefficients of the Bernstein images $Q_{n,m}=B_n[\cos(mx)]$, which we prove using an explicit coefficient formula and a two-regime positivity argument. We also discuss the higher-dimensional sphere analogue and show that the naive affine Bernstein operator fails to preserve the positive-definite cone already on $S^2$.

Reconstructing unknown electromagnetic sources from far-field radiation patterns is a fundamental inverse problem with broad applications in biomedical imaging, non-destructive testing, and telecommunications. In practical settings, however, collecting dense multi-frequency far-field measurements at the Nyquist sampling rate is often infeasible. Under-sampled or sparse data introduce non-radiating source components that sever the uniqueness of the solution, creating severe artifacts when standard inversion techniques are applied. To overcome this limitation, we present a two-stage reconstruction strategy exploiting the physical property that compactly supported, geometrically sparse sources exhibit a finite rate of innovations (FRI). In the first stage, we construct an associated wrap-around structured Hankel matrix. By leveraging the low-rank property of the matrix due to FRI of the unknown sources, we enrich the sub-sampled data. To that end, we convert missing multi-frequency far-field data recovery into a constrained matrix completion task solved via Annihilating Filter-based Low-rank Hankel Matrix Completion Approach (ALOHA). In the second stage, a Fourier inversion scheme reconstructs the current source density from the enriched dataset. Extensive numerical evaluations on electromagnetic source models show that our enrichment framework effectively eliminates under-sampling artifacts and resolves non-uniqueness challenges. The method delivers accurate and stable reconstructions under high sub-sampling rates (e.g., with $30$% to $50$% available samples) and strong noise conditions ($10$ dB SNR), outperforming standard $\ell_1$-compressed sensing baselines.

Let \(X_{1,n},\ldots,X_{d,n}\) be \(n\times n\) random matrices built from independent i.i.d. entry arrays, with centered entries, normalized by \(n^{-1/2}\). We prove that, if every entry law has finite fourth moment, then this tuple converges almost surely strongly in \(*\)-distribution to a free circular family with the matching variances. Equivalently, normalized traces and operator norms converge for every fixed noncommutative \(*\)-polynomial, including polynomials with fixed matrix coefficients. No assumption is imposed on the pseudo-variances of the complex entries. The bounded-entry argument applies the spectrum and moment universality estimates of Brailovskaya and van Handel to all self-adjoint linear pencils. The matching Gaussian pencils are reduced to independent Wigner matrices and identified by Anderson's strong convergence theorem. A fixed-level centered truncation, followed by the Bai–Yin norm bound, transfers the result to finite fourth moments.

Advances in automation and high-throughput experimentation have enabled larger and more complex studies involving many factors and tests, creating a growing demand for computationally effective design construction methods. Efficient experimental design remains a key challenge in this context, creating a need for frameworks that can generate large experiments while preserving orthogonality and minimal aliasing. Unlike existing approaches which struggle with scalability, this work introduces an algorithmic framework for constructing large Orthogonal Minimally Aliased Response Surface (OMARS) designs by enumerating and combining weighing designs, three-level matrices with orthogonal columns and a fixed number of non-zero entries per column. Complete enumerations of weighing designs are achieved for designs with up to 24 tests, covering multiple numbers of factors and weights corresponding to two or three zeros per factor. In addition, a validated partial enumeration procedure and a combination method extend the catalog to substantially larger designs. The combination method enables the construction of OMARS designs for any test size that is a multiple of selected base sizes. This paper thus provides the methodology for generating large catalogs of high-quality OMARS designs, well-suited for high-dimensional screening and response-surface modelling in complex industrial and scientific experiments.

On the $q$-quadratic lattices that carry the Askey–Wilson polynomials, the two roles played by a single Bernstein basis on the linear lattice, namely being a nonnegative partition of unity and carrying an orthogonal connection with the classical families, split between two distinct bases. The affine (spectral) basis retains the Bézier properties. A coprimality argument completely classifies the first-order lattice ladder: its solution space has dimensions two and one in degrees one and two, respectively, and is trivial in every degree at least three for every $q\in(0,1)$. The affine factors do not satisfy the weight-shift mechanism that produces the $q$-Racah connection for the generalized-power basis. The generalized-power basis, built from products of two Askey–Wilson monomials, does not furnish a nonnegative partition of unity in a natural positive parameter region, specified in the text; there the unique normalisation summing to unity has sign-changing coefficients. Nevertheless, its connection coefficients with the Askey–Wilson polynomials are identified completely: they are $q$-Racah polynomials $R_m(\mu(k);\,ad/q,\,bc/q,\,q^{-n-1},\,a/b\,|\,q)$ multiplied by fully explicit prefactors. The proof is bispectral, valid for every degree, and rests on the tridiagonal action of the Askey–Wilson operator on the generalized-power basis, with explicit band coefficients. A scaled limit recovers the Wilson/Racah connection, while a second limit gives the $q$-Hahn polynomial part of the normalised $q$-linear connection coefficients. A numerical case study on the NACA 2412 airfoil illustrates the orthogonal machinery.

We consider the problem of high-dimensional inference with the lasso estimator. Different methods including 'double selection' techniques and multiple versions of the 'debiased lasso' have been proposed for this task with noticeable success. However, most guarantees assume strong hypotheses on the underlying data process and the errors in the linear regression model, such as subgaussian designs and independence between errors and the data itself. We show that 'standardizing' one's dataset – a natural procedure in practical penalized regression – leads to the same results under much weaker hypotheses, paying only a small price for not assuming light tails. The key technical point allowed by this step is exploiting the concentration properties of self-normalized processes. Importantly, we prove our results for two different methods closely related to the 'debiased lasso'. The second method performs valid inference even for a misspecified linear model, under mild sparsity conditions similar to the 'double selection' literature.

We give a finite $n$ enumerative derivation of the local surplus marked point process form of Aldous's critical window limit for the Erdős-Rényi random graph. For $p_n=n^{-1}+\lambda n^{-4/3}$, let $\Xi_n$ place an atom at the rescaled size and surplus of each component. Exact component enumeration yields the limiting factorial correlation densities and the uniform bound \[ \mathbb E[(\Xi_n(K))_q]\le C_K^q e^{-c_Kq^3}, \qquad q\ge1, \] for every compact marked window $K$, uniformly in admissible $n$. The cubic order is optimal for the limiting factorial measures, and the estimate persists after summing over all surpluses on compact size intervals. It yields overcrowding and local exponential moment bounds, quantitative truncation of the finite $n$ Laplace functional expansion, and local point process convergence. Using the Janson-Spencer Palm description and a classical all excess estimate, we also recover ordered $\ell^2$ component size convergence.

We study a two-dimensional semiclassical Schrödinger operator whose potential admits one reflection symmetry. Under a rational independence assumption on the harmonic frequencies and a nonvanishing condition \(a_{30}\ne0\), we show that the first two layers of the quantum Birkhoff normal form, together with the sign of \(a_{30}\) and the transverse data \(\{a_{1,2k}\}_{k\ge1}\), determine the Taylor series of the potential at the bottom of the well. The proof is constructive: the \(\hbar^2\)-layer gives a triangular recursion for odd-degree terms, while the classical layer recovers even-degree terms from their resonant projections.

Let $C$ be a smooth projective curve over $\mathbb C$ and let $E$ be a vector bundle over $C$. Let $\text{Quot}_{r,d}(E)$ denote the Quot scheme which parametrizes quotients of $E$ of rank $r$ and degree $d$. Following Joyce's recipe [Joy21], we introduce Joyce's enumerative invariant for the Quot scheme $\text{Quot}_{r,d}(E)$. The invariant can be viewed as a generalization of the virtual fundamental cycle of the Quot scheme. We evaluate intersection pairings on the Quot scheme $\text{Quot}_{\text{rank}(E)-1,d}(E)$ by computing its invariant explicitly. Following the reformulation of sheaf-theoretic Virasoro constraints in terms of Joyce's vertex algebra framework in [BLM24], we give a proof of the Virasoro constraints for the Quot scheme $\text{Quot}_{r,d}(E)$. With the help of these constraints, we compute the (virtual) intersection numbers of $f$-classes on the Quot schemes.

We promote the $n$-birational motivic homotopy category assignment $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, the birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of schemes in $Sm_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.

We prove that the set $\Pi_{\therefore,\infty}$ of uniform Turán densities of possibly infinite families of $3$-graphs contains a terminal interval: there exists $\delta>0$ such that $[1-\delta,1]\subseteq\Pi_{\therefore,\infty}$. Consequently, $\Pi_{\therefore,\infty}$ has positive Lebesgue measure and Hausdorff dimension $1$.

We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$\beta$-associated signal in Alzheimer's disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $\sigma>0$, the chemotactic velocity is $b_\sigma[v]=\nabla K_\sigma[v]$. For every fixed $\sigma>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.

We propose a phase field model of a macrocracked single crystal in which the crack and the geometrically necessary dislocations descend from a single energy functional. Energy minimization alone then decides dislocation nucleation, through an integral criterion evaluated in closed form along slip chords. The criterion yields a size effect inaccessible to point-wise strength conditions: a grain-size-dependent yield stress. With slip suppressed the model reproduces Griffith fracture; with fracture suppressed, the nucleation load measured by the full non-smooth solver agrees with the closed-form nucleation criterion to four percent. The coupled computations produce a two-stage response: at loads an order of magnitude below cleavage, dislocation bands emitted from the notch tip blunt and shield it, raising the initiation load; once the crack grows, the bands heal; a compact cluster of like-signed dislocations travels with the tip, its canceling partner walls pinned at the grain boundary, and the dissipated fracture resistance equals the elastic one. In the purely energetic, dissipationless limit, emission shields the crack but does not toughen it; toughening requires dissipation, incorporated in the sequel through the threshold resistance to dislocation motion.

Programmable quantum control systems increasingly rely on predictive modules for certification, real-time feedback, and autonomous decision-making. This development raises a fundamental question: can self-analyzing quantum platforms universally predict their own experimental outcomes? Wolpert formalized a general impossibility of universal self-prediction. Here we translate that limitation into an explicit laboratory obstruction that can be realized with finite resources. We consider settings with programmable quantum control in which predictors can be embedded as subroutines within the experiments they analyze. Our diagonal construction uses Kleene's recursion theorem to transform any deterministic bounded-time predictor into a reversible protocol encoding its own specification. The resulting protocol invokes the predictor on that specification and deterministically produces a classical pointer record that contradicts the forecast. For efficient predictors, the compilation has polynomial overhead and admits concrete physical realizations as a fault-tolerant quantum circuit and as a minimal Mach-Zehnder interferometer. These realizations connect computability-theoretic self-reference to programmable quantum hardware. We also introduce and formally define Gödel-safe architectures. These architectures block the forbidden causal path from the protocol description to an actuator that can affect the pointer during the same run. We analyze their implications for real-time quantum error correction, including the resulting expressiveness trade-offs. As quantum control loops grow in computational expressiveness, the limits of self-reference cease to be mere mathematical abstractions and become explicit engineering constraints for the reliable operation of autonomous quantum technologies.

Goodstein's theorem states that a certain sequences based on exponential notation for the natural numbers are always finite. The result is independent of Peano arithmetic and is a prototypical example of a proof of termination by transfinite induction. A variant based instead on the Ackermann function has more recently been proposed by Arai et al., and instead is independent of the more powerful theory ${\sf ATR}_0$. However, this result is contingent on rather elaborate normal forms for natural numbers based on a `sandwiching' procedure. This leaves open both the question of whether the sandwiching procedure can be eliminated while retaining the full strength of the Ackermannian Goodstein principle, and whether other normal forms can lead to non-termination. In this article we settle these questions by showing that any Goodstein process based on the Ackermann function is terminating, and indeed the sandwiching procedure gives rise to Goodstein principles of maximal length. We thus obtain an equivalent principle which does not involve normal forms at all and immediately implies all Ackermannian Goodstein principles that have been considered. Our techniques provide a new approach to termination proofs, where terms in a sequence do not necessarily decrease in complexity, but instead are majorized by some “master” process, already known to be terminating.

Let $A=\{a_{0}, a_{1}, \ldots, a_{k-1}\}$ be a set of $k>7$ integers such that $0=a_{0}<a_1<\cdots<a_{k-1}$ and $\gcd(A)=1$. The set $2^{\wedge}A=\{a+b: a, b\in A, a\neq b\}$ is called the restricted sumsets of $A$. Freiman-Lev conjecture is a well-known conjecture which related to restricted sumsets [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40]. Up to now, Freiman-Lev conjecture is still open for all $a_{k-2}\geqslant 2k-4$ and $a_{k-1}\geqslant 2k-2$. In this paper, we complete the proof of the Freiman-Lev conjecture by resolving this final and most challenging case.

We propose a mathematical model that represents the influence of an external actor (influencer) on a network of actors (influenced). The model is an adaptation of control systems within the family of formulations inspired by the Kuramoto Model of synchronisation. Capturing influence as a capacity to affect the character or behaviour of another, we study an external node's ability to influence the synchronisation of the Kuramoto system while its global behaviour is pulled to the external node's frequency and away from its natural mean frequency. In our work, stochastically generated dynamical weights assigned to the links between the network and the external influencer whereby links from one system to the other are assigned via one-sided heavy tail noise, generated by the Gamma distribution. We perform numerical experiments to examine transition points in the ability of the external node to alter the behaviour of the influenced network; either to disrupt synchronisation or to drive it to collective frequencies determined by the external node. We examine the dependence of transition points on the external node's frequency, where too ambitious a driving frequency fails to influence the system while retaining a synchronised state, and reducing achieves a state of synchronisation at a frequency shifted from the mean natural frequency. We also look at the analytic approximation for the system close to synchronisation and fragmentation to understand its behaviour around this limit.

This paper establishes three main results on the extension dimension of syzygy module categories:(1)we prove that excellent ring extensions preserve extension dimensions of syzygy module categories,including syzygy categories of modules of finite projective dimensions;(2) for cleft extensions, we investigate the behavior of extension dimensions under natural nilpotency and projective conditions;(3)for commutative Artin rings, we establish local-global characterisations for the extension dimension.

This paper develops a prescribed performance leader-following consensus protocol for directed networks under event-based broadcasting. Each node schedules its transmissions using only its current state and its last broadcast value, while followers use held neighbor and leader samples between broadcasts. To accommodate jumps in sampled neighborhood errors caused by asynchronous receptions, a decaying receiver-side performance correction term temporarily modifies the prescribed performance function. For a directed network containing a spanning tree rooted at a time-varying leader with bounded trajectory and derivative, we prove existence and uniqueness of complete non-Zeno solutions, boundedness of all closed-loop signals, and explicit prescribed bounds for the sampled and continuous neighborhood disagreements and the leader-following error. These bounds quantify the effects of the communication topology and broadcast thresholds. Simulations verify the prescribed performance bounds and illustrate the event-based communication mechanism.

Fix a positive integer $k\ge2$, and for $n\ge k$, consider a row of $n$ molecules. From among the $n-k+1$ nearest-neighbor $k$-tuples of molecules, select one uniformly at random and bond the $k$ molecules. Now, from all the remaining nearest-neighbor $k$-tuples, again select one uniformly at random and bond the $k$ molecules. Continue like this until there are no nearest-neighbor $k$-tuples left. Let $M^{(n)}_k$ denote the expected value of the number of bonded molecules. An explicit integral formula for $m_k:=\lim_{n\to\infty}\frac{M^{(n)}_k}n$ is known, and an explicit formula for $m_\infty:=\lim_{k\to\infty}m_k$ is known. The constant $m_\infty$, known as the Rényi parking constant, arises as the limiting packing density for a continuous analog of the above discrete packing problems. These are all models of what is called random sequential adsorption (RSA). The first part of this paper studies the gaps of sizes $0,1,\cdots, k-1$ that arise between bonded $k$-tuples and shows that after scaling the $k$-grid, when $k\to\infty$ the empirical distribution of expected gaps in the discrete problem on the lattice converges weakly to an appropriate gap distribution that is known to hold for the above noted continuous analog. The second part of the this paper considers two different models of the discrete bonding problem when both $k_1$-bonding and $k_2$-bonding occur, with $2\le k_1<k_2$. Explicit formulas are obtained for the analogs of $m_k$, and the asymptotic behavior of these analogs is studied both when $k_2\to\infty$ with $k_1$ fixed, and when $k_1,k_2\to\infty$ at certain ratios.

Let \(X\) be a closed Riemann surface and let \(q\in H^0(X,K^4)\) be a nonzero holomorphic quartic differential on \(X\). For \(t >0\), the ray \(tq\) determines a family of Hitchin representations in the \(\operatorname{PSp}(4,\mathbb R)\)-Hitchin component. We study, as \(t\to+\infty\), the asymptotic behavior of their holonomy along closed curves. We obtain explicit asymptotic formulas for all singular values and for the absolute values of all eigenvalues of the holonomy. Their logarithmic growth rates are given by integrating the local fourth roots of \(q\) along the saddle connections forming the geodesic representative of the curve with respect to the singular flat metric \(\lvert q\rvert^{1/2}\). No restriction is imposed on the orders of the zeros of \(q\).

In this paper, we introduce the Bernoulli–Strang–Fix conditions and their generalized versions for a vector-valued generator $\varphi=(\varphi_0,\dots,\varphi_{\rho-1})$ and a periodic nonuniform sampling set $X$. We use these conditions to establish exact and asymptotic polynomial reproduction properties of sampling Kantorovich operators associated with $(\varphi,X)$ up to a prescribed degree. We analyze the approximation properties and convergence behavior of these operators in detail. Furthermore, we demonstrate their application to signal prediction from a finite number of past local average samples, showing that sampling Kantorovich operators can also serve as effective prediction operators. Finally, we present numerical examples based on Gaussian functions and B-splines to illustrate and validate the theoretical approximation and prediction results.

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch–Kronheimer–Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal–Szekeres spacetime in terms of its radial null geodesics.

We study the quantization of algebraic varieties defined by equations involving Casimir polynomials of compact semisimple Lie algebras. The Casimir polynomials belong to the Poisson center of the corresponding Lie-Poisson algebra. For this purpose, we employ a recently developed matrix regularization of Lie-Poisson algebras. In particular, using its formulation based on reducible representations, we construct quantizations of these algebraic varieties through their decomposition into coadjoint orbits, including singular orbits. As a concrete example, we present the construction of fuzzy $S^7$ in detail.

Homogeneous Crump-Mode-Jagers processes is considered with arbitrary initial population. The asymptotic distribution of the extinction time in the limit of large initial population is obtained in subcritical regime by assuming the existence of Malthusian parameter.

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $\nu_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson–Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo–Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.

We propose the stochastic block Ornstein-Uhlenbeck (SBOU) process, a continuous-time multivariate model in which the drift matrix encodes a latent group structure among its components. Our main contribution is a community-detection algorithm whose misclassification proportion converges to zero in a regime combining infill, long-span, and high-dimensional asymptotics. To our knowledge, this is the first consistency result of this kind for latent group recovery in a discretely observed continuous-time multivariate model. As a key intermediate result, we establish consistency of the discretely observed maximum likelihood estimator of the drift matrix in the same regime, thereby extending the high-dimensional Lévy-driven Ornstein-Uhlenbeck literature. For practical implementation, we develop a feasible model-selection procedure for estimating the support of the drift matrix, which enables data-driven selection of the number of latent groups. The SBOU framework can be viewed as a continuous-time generalisation of the discrete-time stochastic-block VAR model, allowing for both positive and negative dynamic interactions between groups as opposed to only positive. We illustrate the methodology on the RE-Europe wind-capacity dataset and recover a country-level grouping consistent with the geographic benchmark.

We calculate the Batalin-Vilkovisky (BV) algebra structure of Rabinowitz loop homology for compact rank one symmetric spaces. As a consequence, we prove that such a space satisfies a natural homological condition called string point invertibility if and only if its Euler characteristic is equal to the characteristic of the coefficient field for loop homology. This implies certain cases of Viterbo's conjecture on a uniform bound on the spectral norm of exact Lagrangian submanifolds in cotangent disk bundles. Furthermore, we prove that whenever a compact rank one symmetric space is string point invertible, the critical levels of its loop homology classes with respect to an arbitrary Riemannian metric satisfy a resonance condition with respect to degrees and a density condition for closed geodesics. This generalizes results of Hingston and Rademacher for spheres to a broader class of compact rank one symmetric spaces.

We characterize smooth proper Deligne-Mumford stacks $\mathscr{X}$ that arise as compactifications of ball quotient stacks $[\mathbb{B}^d/\Gamma]$. Moreover, we show that every ball quotient admits a compactification whose boundary divisor $\mathscr{D}:=\mathscr{X}-[\mathbb{B}^d/\Gamma]$ is a disjoint union of quotient stacks $[A/G]$, where $A$ is an abelian variety and $G$ is a finite group. This generalizes a result of Deng-Cadorel. Our strategy combines Simpson's non-abelian Hodge correspondence for smooth proper DM-stacks, Mochizuki's generalization of the classical Simpson's correspondence to the log setting, and the uniformization results of Deng-Cadorel.

Let $A$ be a finite dimensional algebra. The lattice of torsion pairs in $\rm mod (A)$ is controlled by cosilting pairs, infinitely generated analogues of support $\tau^-$-tilting pairs. Then, edges in the Hasse quiver (i.e. minimal inclusions of torsion-free classes) correspond to irreducible mutations of cosilting pairs. An important difference with classical $\tau$-tilting theory is that not all indecomposable summands of a cosilting pair are mutable. So, it is very important to identify mutable indecomposable summands in a given cosilting pair. It is well-known that mutable summands correspond to injective envelopes of finitely presented simples in the HRS-tilted heart. Based on this correspondence, we first present a method for obtaining all simples in the HRS-tilted heart, and then give some necessary and sufficient conditions for an indecomposable summand of a given cosilting pair to be left mutable or right mutable.